the anisotropic hyperelastic biomechanical response of the vocal

Purdue UniversityPurdue e-PubsSchool of Mechanical Engineering FacultyPublications School of Mechanical Engineering

2013

The anisotropic hyperelastic biomechanicalresponse of the vocal ligament and implications forfrequency regulation: A case studyJordan E. KelleherPurdue University

Thomas SiegmundPurdue University, [email protected]

Mindy DuUniversity of Texas Southwestern Medical Center

Elhum NaseriUniversity of Texas Southwestern Medical Center

Roger W. ChanUniversity of Texas Southwestern Medical Center

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Recommended CitationKelleher, Jordan E.; Siegmund, Thomas; Du, Mindy; Naseri, Elhum; and Chan, Roger W., "The anisotropic hyperelastic biomechanicalresponse of the vocal ligament and implications for frequency regulation: A case study" (2013). School of Mechanical EngineeringFaculty Publications. Paper 11.http://docs.lib.purdue.edu/mepubs/11

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The anisotropic hyperelastic biomechanical response of the vocalligament and implications for frequency regulation: A case study

Jordan E. Kelleher and Thomas Siegmunda)

Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, Indiana 47907

Mindy Du, Elhum Naseri, and Roger W. Chanb)

Otolaryngology-Head and Neck Surgery, University of Texas Southwestern Medical Center,5323 Harry Hines Boulevard, Dallas, Texas 75390

(Received 30 June 2012; revised 12 November 2012; accepted 31 December 2012)

One of the primary mechanisms to vary one’s vocal frequency is through vocal fold length changes.

As stress and deformation are linked to each other, it is hypothesized that the anisotropy in the

biomechanical properties of the vocal fold tissue would affect the phonation characteristics. A

biomechanical model of vibrational frequency rise during vocal fold elongation is developed which

combines an advanced biomechanical characterization protocol of the vocal fold tissue with

continuum beam models. Biomechanical response of the tissue is related to a microstructurally

informed, anisotropic, nonlinear hyperelastic constitutive model. A microstructural characteristic

(the dispersion of collagen) was represented through a statistical orientation function acquired from

a second harmonic generation image of the vocal ligament. Continuum models of vibration were

constructed based upon Euler–Bernoulli and Timoshenko beam theories, and applied to the study

of the vibration of a vocal ligament specimen. From the natural frequency predictions in

dependence of elongation, two competing processes in frequency control emerged, i.e., the applied

tension raises the frequency while simultaneously shear deformation lowers the frequency. Shear

becomes much more substantial at higher modes of vibration and for highly anisotropic tissues.

The analysis was developed as a case study based on a human vocal ligament specimen.VC 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4776204]

PACS number(s): 43.70.Bk, 43.70.Aj, 43.70.Fq [BHS] Pages: 1625–1636

I. INTRODUCTION

The human voice has the capability to achieve a funda-

mental frequency F0 range of five octaves.1 Many mecha-

nisms contribute to sound production of the voice and F0

control. Following the myoelastic-aerodynamic theory of

phonation, these mechanisms are associated with the biome-

chanical properties of the vocal folds and the aerodynamic

driving forces via the subglottal pressure.2 The biomechani-

cal characteristics of the vocal folds can be categorized into

passive (e.g., mass, length, elasticity) and active features

(e.g., isotonic and isometric muscle contraction), which are

inter-dependent, to regulate F0.3 The vocal ligament (middle

and deep layers of the lamina propria) is considered to be the

primary load-bearing portion of the vocal fold, especially at

high longitudinal stretches. It is part of the vocal fold struc-

ture that is the focus of the present study. Therefore, the

vocal ligament has a critical functional role in phonation-

enabling high vocal pitches by supporting large longitudinal

stresses.

The string model of phonation is often used to model

the change in F0 due to variation of length and tension. Mod-

eling experimental results of the vocal folds with the ideal

string law dates back at least to the mid-20th century.4 While

the string model of phonation has widely been used to esti-

mate the natural frequencies of phonation, such a model

ignores the bending stiffness (flexural rigidity) of the vocal

fold structure and thus underestimates frequencies.5 Beam

models inherently account for bending stiffness allowing the

prediction of a non-zero F0 even in an unstretched state

(with no applied tension), which is not possible in the string

model.6,7 The nonlinear biomechanical properties of the

vocal folds have been integrated in F0 predictions using

string and classical (i.e., Euler–Bernoulli) beam models.5,7,8

The Euler–Bernoulli beam model is appropriate for struc-

tures with high length-to-thickness ratios (> 100) and for

isotropic or nearly isotropic materials. Such conditions are

typically not met in vocal folds.

Recent clinical studies have measured the vocal fold

elongation using high-speed endoscopy and the acoustic sig-

nal (i.e., F0) during a glissando in professional singers9,10

and non-singers.11 Older techniques have used photography

to inspect the vocal fold membranous length during an

arpeggio.12–14 Still others have utilized radiographic meth-

ods such as traditional x-ray imaging15 or computed tomog-

raphy.16 Finally, vocal fold elongation during frequency rise

has been investigated in vitro with human larynges.4

The extracellular matrix of the vocal fold lamina propria

is comprised of fibrous (e.g., collagen and elastin) and inter-

stitial proteins (e.g., proteoglycans). These proteins are

attributed to providing the elastic and viscous properties

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected])Also at: Biomedical Engineering, University of Texas Southwestern Medi-

cal Center, 5323 Harry Hines Boulevard, Dallas, Texas 75390.

J. Acoust. Soc. Am. 133 (3), March 2013 VC 2013 Acoustical Society of America 16250001-4966/2013/133(3)/1625/12/$30.00

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.210.206.145 On: Mon, 01 Feb 2016 15:29:55

mailto:[email protected]

necessary to sustain the stresses and strains during phona-

tion.17 A microstructural arrangement of elastin and collagen

fibers aligned primarily in the anterior-posterior direction

has been demonstrated in histological studies.18–20 The ani-

sotropy of the vocal fold tissue was also confirmed in meas-

urements of the biomechanical properties. A protocol was

recently developed for the characterization of the tissue ani-

sotropy in the small strain regime by combining longitudinal

stretch experiments with transverse indentation experi-

ments.21 The anisotropy, expressed as the ratio of the

anterior-posterior elastic modulus and the longitudinal shear,

was found to range typically from 15 to 40, while for an

incompressible, isotropic tissue the ratio would equal three.

As it was shown in Kelleher et al.,22,23 the effect of shear de-

formation can be of substantial magnitude in the vibration of

vocal fold tissue, and thus seems crucial for accurate predic-

tions of the fundamental frequency. Furthermore, it is under-

stood that during the stretching of a tissue (i.e., during

posturing of the vocal folds) a re-orientation of the fibrous

proteins can occur which will alter the effective anisotropy.

Based on these findings, it is argued here that vocal fold

tissue should be described as a nonlinear and anisotropic

solid. It is then hypothesized that shear deformation would

play a significant role in vocal fold vibration. A structural

model for vocal fold vibration accounting for longitudinal

and shear stiffnesses is thus employed to provide relevant

insight into vocal fold vibration during phonation.

In the present study, the vocal ligament is characterized

by a microstructurally informed, anisotropic, nonlinear,

hyperelastic constitutive model, specifically the Gasser–

Ogden–Holzapfel (GOH) model.24 The GOH model is a his-

tomechanical model with constitutive parameters obtained

from the tissue’s histology and from biomechanical experi-

ments. For the present investigation, the microstructural data

was obtained using multiphoton microscopy techniques, see

Miri et al.25 for a recent application of this technique to

vocal fold tissue. Multiphoton microscopy has emerged as a

powerful tool for imaging a tissue’s collagen fiber architec-

ture without using exogenous stains.26 In order to accom-

plish this non-linear excitation, highly focused femtosecond

pulsed lasers are typically used. Since the laser source is in

the infrared range, the scattered light is minimized allowing

for relatively deep imaging into the tissue (approximately

500 lm) while limiting the adverse effects of photobleach-

ing.27 The biomechanical experiments commonly employed

are uniaxial or biaxial stretch experiments. Then, the biome-

chanical parameters of the GOH model are inversely fitted to

the experimental test data. This approach commonly causes

uncertainty in the accuracy of the model predictions in the

small strain regime. Here, this problem is overcome by sup-

plementing the information from the stretch experiment in

the longitudinal direction of the vocal folds with data from a

transverse indentation experiment.21

The structural model of phonation considers the vocal

ligament as a beam. The geometry of the vocal ligament

may be approximated as a beam, though the beam represen-

tation may not be suitable for the vocal fold structure as a

whole (cover, ligament, and muscle). For beams with small

length-to-thickness ratios and/or relatively high levels of

anisotropy (longitudinal elastic modulus to longitudinal

shear modulus ratio > 10), the Timoshenko beam theory is

appropriate.28 Timoshenko beam theory allows the beam’s

cross-section to rotate with respect to the neutral axis; hence,

two additional effects are included: shear deformation and

rotary inertia. Therefore, with longitudinal stretch during

vocal fold posturing, the vibration characteristics will

depend on the current longitudinal tangent stiffness and the

longitudinal shear stiffness in the stretched state.

The purpose of this study was to develop a microstruc-

tural constitutive model of vocal ligament tissue deforma-

tion which is then linked to a beam model for computation

of the modal frequencies. An experimental protocol for tis-

sue testing and microscopy techniques is defined. To exem-

plify the proposed approach, the protocol is applied to an

investigation of a vocal ligament specimen to obtain biome-

chanical and microstructural parameters. Bounds of the tis-

sue constitutive model are evaluated and the vibration

response of these limits is discussed. The present study is

developed in the context of a case study. Thereby, one spe-

cific vocal ligament specimen excised from a human larynx

is considered.

II. MATERIALS AND METHODS

A. Materials

The specimen under consideration was a vocal ligament

carefully dissected from an excised 45 year-old Caucasian

male larynx. The larynx was procured from the Willed Body

Program of the University of Texas Southwestern Medical

Center approximately 18 hours postmortem, and gross exam-

ination of the vocal folds revealed no abnormalities or path-

ologies. The sample preparation and testing protocols were

approved by the Institutional Review Board of University of

Texas Southwestern Medical Center. The ligament specimen

was dissected with portions of the thyroid and arytenoid car-

tilages remaining to maintain the natural anterior and poste-

rior attachments. The ligament specimen was separated from

the underlying vocalis muscle and immediately placed in

phosphate buffered saline (PBS).

B. Biomechanical testing

Upon dissection, biomechanical testing was conducted

at the University of Texas South-western Medical Center.

The vocal ligament specimen was subjected to longitudinal

tension in the anterior-posterior direction following the

approach described in detail in Kelleher et al.23 The dis-

placements of two points on the specimen surface at the an-

terior commissure and vocal process were measured

optically by use of a CCD camera system and an image anal-

ysis program. Therefore, the stretch k of the ligament speci-

men due to elongation DL and with an initial length L0, was

calculated as k¼ 1þDL/L0. Distance measurements were

calibrated by taking an image of an object of known dimen-

sions to establish a pixel-to-mm ratio. The specimen diame-

ter of the entire tissue specimen was optically measured at

thirteen equidistant locations, and the cross-sectional area

was calculated assuming the tissue specimen to be of circular

1626 J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013 Kelleher et al.: Microstructural model and frequency regulation


cross-section. A second CCD camera, orthogonal to the first,

confirmed the circular cross-section to be a reasonable

assumption (the diameter measurements from each camera

were within 10% on average). Then the thirteen area meas-

urements were averaged to estimate the initial cross-

sectional area A0. Finally, the nominal stress rN¼P/A0 was

determined from the force output P and the specimen’s

cross-sectional area A0 in the undeformed state. After the

stretch experiments, the specimen was placed in PBS and

allowed to rest for approximately 2 h.

Subsequently, a transverse (i.e., medial-lateral direc-

tion) indentation experiment was also performed on the

vocal ligament specimen in a state where longitudinal

stretch was absent.21 The compressive force-displacement

response was evaluated using an anisotropic contact

model.29 The force-displacement relation for an anisotropic

half-space contacted by a rigid frictionless sphere, which

leads to the same solution for two contacting cylinders with

perpendicular axes, is

P0 ¼ 4M

ffiffiffiRp

3d3=2; (1)

where P0

is the indentation force, R is the radius of curva-

ture, d is the indentation depth, and M is the indentation

modulus. The indentation modulus is an integral function

of components of the elastic stiffness tensor, considering

the ligament to be transversely isotropic (see Appendix A).

For a known longitudinal elastic modulus (measured from

the tensile test), the transverse elastic modulus and longitu-

dinal shear modulus were obtained by inversely fitting Eq.

(1) to the experimental P0-d curve.21 An estimate of the

“degree of anisotropy” in the unstretched state was defined

as the ratio of the longitudinal elastic modulus to the longi-

tudinal shear modulus. For purposes of estimating the ani-

sotropy only, the longitudinal elastic modulus was

computed by measuring the localized stretch of the mid-

membranous region, which was where the indentation was

performed. Since vocal fold tissue has been shown to be

heterogeneous, it was important that the elastic properties

be measured in the same region.22,30 However, for the con-

stitutive and vibration modeling the elastic properties of the

overall tissue (i.e., from the anterior commissure to the

vocal process) were used.

C. Microscopy and image processing

In order to utilize a microstructurally informed constitu-

tive model, knowledge of the tissue’s micro-architecture

(specifically the directionality and dispersion of collagen

fibers) is needed. Immediately following mechanical testing,

the vocal ligament specimen was embedded in Optimal Cut-

ting Temperature (Tissue-TekVR

, Sakura Finetek, Inc., Tor-

rance, CA) compound, where the medial-lateral and

anterior-posterior directions were labeled, and placed in a

�20 �C freezer until sectioning. Once the specimen was fro-

zen, it was transferred to a cryostat (i.e., a microtome in an

enclosure maintained near �20 �C) and sectioned into 10 lm

slices in the sagittal plane. The section taken from the mid-

plane between the medial and lateral surfaces was placed

onto a glass slide for viewing under a microscope. Figure 1

provides a depiction of the location and orientation of the

section used for imaging.

Microscopy was performed at the Live Cell Imaging Fa-

cility at University of Texas Southwestern Medical Center.

Microstructural images of the vocal ligament specimen were

acquired using a Zeiss laser scanning confocal microscope

510 META using an Achroplan water immersion objective

with 40� magnification, 0.8 numerical aperture, and a work-

ing distance of 3.6 mm. Excitation was achieved using a tun-

able (705–980 nm) coherent Chameleon Ti:Sapphire pulsed

near-infrared laser at an average power of 1.3 W. The excita-

tion wavelength in this study was 900 nm. The resulting

512� 512 pixel image had a field of view of 230� 230 lm,

which revealed the micro-architecture of the unstained sam-

ples. The backward (i.e., reflected light) and forward (i.e.,

transmitted light) second harmonic generation (SHG) signals

of the microstructure of the vocal ligament specimen were

simultaneously detected in two channels. The reflected

image was not considered in this investigation because it

produced a weaker signal than the transmitted image. Thus,

only the transmitted image was analyzed for resolving the

microstructural arrangement.

The raw SHG image from the mid-membranous location

(mid-coronal plane) was loaded into MATLABVR

(The Math-

works, Natick, MA) and analyzed using an automated,

custom-programed script to determine the distribution of

fiber orientation. Initially, the image was contrast enhanced

such that 1% of data was saturated at low and high inten-

sities of the original image, and then padded with the mean

gray scale value increasing the image size to 1024� 1024

pixels. To reduce edge effects in the spectral analysis, a two-

dimensional Hanning window was applied. The two-

dimensional discrete Fourier transform was implemented

and the power spectrum was displayed.31 The fiber align-

ment was calculated by analyzing the pixel intensity values

of the power spectrum in polar coordinates, while omitting

the pixels with a radial component less than 5 pixels to avoid

erroneous low frequency effects. The relative intensity (RI)

in angular increments of 62� was

FIG. 1. The location and orientation of the vocal ligament section used in

the microscopy. a-p is the anterior-posterior, m-l is the medial-lateral, and

i-s is the inferior-superior direction, respectively.

J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013 Kelleher et al.: Microstructural model and frequency regulation 1627


RIðHÞ ¼

Xh<Hþ2

h�H�2

gðr; hÞ

Xh<180�

h�0�gðr; hÞ

; (2)

where g is the intensity value of the pixel with polar coordi-

nates (r, h) in the power spectrum.32 Two post-processing

steps were performed on RIðHÞ before the preferred fiber

orientation was determined. First, RIðHÞ was shifted such

that the maximum value was now centered at H ¼ 0�. This

step was necessary in order to fit the data to a statistical

(e.g., von Mises) distribution. Finally, the minimum value of

the intensities was subtracted off of all RIðHÞ to remove the

“dc” or baseline component of the grayscale intensities.

From the power spectrum, a parameter was extracted

which characterized the microstructure. A single population

of fibrous proteins is assumed, aligned in a transversely iso-

tropic orientation while allowing for dispersion around the

preferred/main axis of orientation. Following an approach

suggested in Gasser et al.,24 the RIðHÞ from the SHG image

was fitted to a von Mises distribution to find the density dis-

tribution function qðHÞ:

qðHÞ ¼ 2qðHÞðp

0

qðHÞsinH dH;

qðHÞ ¼ A exp½b cos ð2HÞ�

2

ðp

0

exp ½b cos H� dH; (3)

where A and b are fitted parameters of the von Mises distri-

bution. The dispersion of the collagen fibers was subse-

quently quantified by using the parameter j which represents

the degree of fiber dispersion around the preferred anterior-

posterior direction:

j ¼ 1

4

ðp

0

qðHÞsin3H dH: (4)

The fiber dispersion coefficient ranges from j¼ 0 for per-

fectly aligned fibers to j¼ 1/3 for randomly distributed

fibers (i.e., isotropic).

D. Microstructural constitutive model

The constitutive model employed for this investigation

is the Gasser–Ogden–Holzapfel (GOH) model.24 The GOH

model regards the tissue’s mechanical response as a superpo-

sition of contributions from ground substances and from

fibers. The interstitial proteins (“ground substances”) were

modeled with an incompressible, isotropic, neo-Hookean

model, and the fibrous proteins were modeled by an expo-

nential, anisotropic, hyperelastic stress potential.

The GOH model was solved for the case of uniaxial ten-

sile stretch k along with longitudinal shear strain c simulta-

neously. The longitudinal shear strain was included in the

prescribed deformation so that an expression of the shear

modulus as a function of k may be obtained. The actual

value of c was set to zero so that no shear strain was present

in the model, but derivatives with respect to c are available.

For tensile stretch k applied in the longitudinal direction

(treated as the x-axis or 11 subscript) and a longitudinal

shear strain c, the resulting Cauchy stress in the anterior-

posterior direction r11 is

r11 ¼2

3ð~s11 � ~s22Þ;

~s11 ¼ cðk2 þ c2Þ þ 2wf ½k2ð1� 2jÞ þ jc2�;~s22 ¼ ðcþ 2jwf Þ=k;wf ¼ k1eexpðk2e2Þ;e ¼ k2ð1� 2jÞ þ jc2 � 1þ 2j=k: (5)

The detailed algebraic and matrix manipulations leading to

Eq. (5) are given in Appendix B. The mechanical response

of Eq. (5) is described by four parameters: the isotropic neo-

Hookean parameter c, a modulus-like parameter k1, a dimen-

sionless parameter indicative of the degree of nonlinearity

k2, and a microstructural parameter j representing the colla-

gen fiber dispersion. In order to compare to the experimental

stress-stretch data, the nominal stress (which was equivalent

to the first Piola–Kirchhoff stress in this case) is defined as

rN11 ¼

2

3kð~s11 � ~s22Þ: (6)

The tangent longitudinal elastic modulus (i.e., the in-

stantaneous tensile stiffness at a given level of stretch) was

obtained by partial differentiation of the Cauchy stress r11

with respect to k. Partial differentiation was necessary

because r11 was a function of two variables: k and c. The

tangent elastic modulus Et is

Et ¼@r11

@k¼ 2

3

@~s11

@k� @~s22

@k

� �;

Et ¼4

3ckþ

@wf

@kk2ð1� 2jÞ � j

kþ jc2

� ��

þ 2wf kð1� 2jÞ þ 1

2k2ðcþ 2wf jÞ

�;

@wf

@k¼ 2k1 kð1� 2jÞ � j

k2

� �expðk2e2Þ þ k2wf

@ðe2Þ@k

:

(7)

The tangent longitudinal shear modulus Gt as a function of kis obtained by differentiating the longitudinal Cauchy shear

stress r13 with respect to the shear strain as

Gt ¼@r13

@c¼ 1ffiffiffi

kp cþ 2jc

@wf

@cþ 2jwf

� �;

@wf

@c¼ 2jk1cexpðk2e2Þ þ 4jk2cwf

� jc2 þ k2ð1� 2jÞ þ 2jk� 1

� �: (8)

Considering the unstretched state (k¼ 1, c¼ 0) and

known ratio Et,0/Gt,0, the number of parameters for the



GOH model reduce to two as the following expression

emerges:

Et;0

Gt;0ðk ¼ 1; c ¼ 0Þ ¼ 2þ 8k1

3cð1� 3jÞ2: (9)

The experimental stress-stretch response was fit to the

expression of Eq. (6) using an optimization process executed

with the Global Optimization ToolboxTM

of MATLABVR

. The

algorithm solves the least-squares problem for multiple start

points of k1 and k2 in an effort to find the global minimum

rather than local minima. Constraints were imposed on the

parametric search space so that the upper bounds of k1 and

k2 were [1000 kPa, 1000], respectively, and the lower

bounds were zero. 50 start points were randomly selected

over the range of possible values in the optimization process.

The set of k1 and k2 that was deemed the optimal solution

was one that minimized the residual sum of squares (RSS).

As a measure of the goodness-of-fit, the coefficient of deter-

mination R2¼ 1�RSS/TSS, where TSS is the total sum of

squares, was calculated.

For comparison and discussion purposes, two hypotheti-

cal constitutive characterization cases are considered: (1) a

case assuming initial isotropic response and (2) a case

assuming near-perfect microstructural alignment. For the

ideal isotropic case (j¼ 1/3), the measured anisotropy in the

unstretched state Et,0/Gt,0 cannot be imposed as a constraint,

and therefore an approximate isotropic case is considered

with j¼ 0.3. For j¼ 0 (perfectly aligned fibers), the equa-

tions significantly reduce in complexity. In particular, the

tangent shear modulus becomes Gt ¼ c=ffiffiffikp

which indicates

the absence of load transfer across the fibers in shear. A

near-perfect microstructure alignment with j¼ 0.01 will be

used instead.

E. Vibration-dynamic beam models

The vocal ligament vibration characteristics were

derived from the analysis by treating the ligament structure

as a beam with circular cross-sections and an incompressi-

ble, anisotropic elastic solid. At each level of applied longi-

tudinal stretch, the tangent stiffnesses Et and Gt were

computed and subsequently inserted into the beam model

equations. The length, diameter, cross-sectional area, and

second moment of inertia in the current/deformed configura-

tion, L ¼ kL0; D ¼ D0=ffiffiffikp

; A ¼ A0=ffiffiffikp

; and I ¼ 0:25pðD=2Þ4 (assuming incompressibility), respectively, were

used rather than the quantities in the reference/undeformed

configuration, as in Zhang et al.7

For an Euler–Bernoulli beam subjected to a longitudinal

tensile load, the governing equation is

EtI@4w

@x4� P

@2w

@x2þ qtA

@2w

@t2¼ 0; (10)

where w is the deflection of a beam segment located a dis-

tance x from the anterior end, t is time, P is the applied lon-

gitudinal tension which is P¼r11 A, and qt is the tissue

density.33 The first term on the left hand side of Eq. (10) con-

stitutes the bending resistance (flexural rigidity) of the beam,

the second term accounts for the applied tension, and the last

term is the inertial effects of the lateral (translational) dis-

placement. The solution can be written as a linear superposi-

tion of four terms (sine, cosine, hyperbolic sine, and

hyperbolic cosine) along with arbitrary constants to be deter-

mined by the boundary conditions. Therefore, the frequency

of the nth mode for a pinned-pinned Euler–Bernoulli beam

as a function of the longitudinal stretch is34

FEBn ðkÞ ¼

ðnpÞ2

L2

ffiffiffiffiffiffiffiEtI

qtA

s1þ PL2

ðnpÞ2EtI

" #1=2

: (11)

For a Timoshenko beam under a longitudinal tensile

load, the two governing equations are35

1þ P

ksGtA

� �EtI

@4w

@x4� P

@2w

@x2þ qtA

@2w

@t2

� qtI 1þ P

ksGtAþ Et

ksGt

� �@4w

@x2@t2¼ 0;

1þ P

ksGtA

� �EtI

@4/@x4� P

@2/@x2þ qtA

@2/@t2

� qtI 1þ P

ksGtAþ Et

ksGt

� �@4/@x2@t2

¼ 0; (12)

where / is the rotation of the beam’s cross-section and ks is

Timoshenko’s shear coefficient. In Eq. (12), shear deforma-

tion and rotary inertia are accounted for. The last term on the

left hand side constitutes the joint action of shear deforma-

tion and rotary inertia but is of negligible proportions,

according to Abramovich,35 so it is omitted in the solution

process. After the separation of variables method and apply-

ing boundary conditions, the nth mode frequency variation

with applied stretch of a pinned-pinned Timoshenko beam is

FTn ðkÞ ¼

ðnpÞ2

L2

ffiffiffiffiffiffiffiEtI

qtA

s1

b1þ PL2

ðnpÞ2EtIþ P

ksGtA

!" #1=2

;

b ¼ 1þ ðnpÞ2 I

AL21þ P

ksGtA

� �þ EtI

ksGtAL2

:

(13)

Timoshenko’s shear (correction) coefficient ks is a constant

which depends upon the shape of the beam’s cross-section.36

For a circular cross-section and incompressible tissue

ks¼ 0.964. It is important to note that the beam models

developed here reveal the modal frequencies of the vocal lig-

ament tissue and do not encompass flow-induced, self-sus-

tained vibration.

III. RESULTS

The SHG microstructural image and the results of

subsequent image processing steps for the specimen of the

present case study are displayed in Fig. 2. The image

revealed the extracellular matrix fibrous proteins, Fig. 2(a).

A waviness or crimp typical of collagen fibers is evident.

The main fiber direction, assumed to be associated with

the anterior-posterior direction, was approximately 70�



counter-clockwise from the SHG image’s horizontal axis

(i.e., abscissa). Figure 2(b) depicts the relative intensity dis-

tribution, shifted such that the maximum was centered at

H ¼ 0� (i.e., the anterior-posterior axis). The von Mises den-

sity distribution fit, which has a goodness-of-fit R2¼ 0.88, is

also shown. The value of the fiber dispersion coefficient

computed from this particular ligament’s microstructure was

j¼ 0.1444. Thus, the ligament specimen of the present case

study displayed a microstructure that is strongly aligned but

possesses significant dispersion. This level of dispersion was

found to be representative for the mid-membranous region

of the vocal ligament.37

The vocal ligament specimen exhibited a time depend-

ent response with the stress at maximum load declining con-

tinuously with the number of load cycles in both the stretch

and the indentation experiments. Even after 180 cycles, no

steady state response was reached, consistent with other

studies.23,38 Thus, the analysis considers one representative

cycle such that transient effects associated with cyclic load-

ing were minimized. In particular, data is presented for the

55th loading cycle of the stretch and indentation tests. From

cycles 1 to 180 in the stretch test, the peak force decayed by

33%. From cycles 1 to 60 in the indentation test, the peak

force decayed by 25%. The detailed analysis of the ligament

specimen was conducted for the 55th cycle where the peak

force had experienced 76 and 99% of the total decay for the

stretch and indentation tests, respectively.

The tensile stress-stretch curve for the overall tissue

(i.e., from anterior commissure to the vocal process) of the

55th cycle is depicted in Fig. 3(a) with the rN-k response

being highly nonlinear and hysteretic. The tensile response

of the middle segment (i.e., mid-membranous region) exhib-

ited an initial longitudinal elastic modulus of 28 kPa. Analy-

sis of the transverse indentation experiment [see Fig. 3(b)]

revealed that the transverse elastic modulus was 2.5 kPa and

the longitudinal shear modulus was 0.9 kPa. Therefore, Et,0/

Gt,0¼ 25 was used to approximate the degree of anisotropy

in the unstretched state for this ligament specimen, which is

a typical value for the vocal ligament.21 The modeled curve

in Fig. 3(b) was fitted to only the initial 0.3 mm of the inden-

tation depth due to the transverse indentation model’s

restriction to the small strain regime.21

The GOH model with constitutive parameters

k1¼ 8.14 kPa, k2¼ 27.32 for j¼ 0.1444 and c¼ 0.30 kPa

FIG. 2. Microscopy and digital image processing.

FIG. 3. The biomechanical stretch and indentation experiments.



was found to approximate the experimental stress-stretch

curve well (R2¼ 0.8360), Fig. 3(a). It should be noted again

that these parameters were determined under constraint of

Eq. (9). Figure 3(a) depicts the outcome of the constitutive

description by assuming a priori a near perfectly aligned

microstructure (j¼ 0.01) together with the measured anisot-

ropy ratio. The constitutive parameters for this approxima-

tion were k1¼ 2.85 kPa, k2¼ 11.18, and c¼ 0.31 kPa. Again,

a good description of the uniaxial stretch data was achieved,

R2¼ 0.8359. Finally, assuming a nearly isotropic (j¼ 0.3)

tissue and Et,0/Gt,0¼ 3 representative of the isotropic

response, the parameters were k1¼ 119.89 kPa, k2¼ 243.65,

and c¼ 3.20 kPa (R2¼ 0.8362). It should be noted that the

cases of j¼ 0.01 and j¼ 0.3 are introduced for comparison

only and do not represent the actual tissue specimen in this

case study.

Figure 4(a) shows the evolution of the degree of anisot-

ropy Et/Gt with longitudinal stretch, as predicted by the

GOH model for three values of fiber dispersion: j¼ 0.1444

(the measured value of fiber dispersion), j¼ 0.01 (virtually

no fiber dispersion), and j¼ 0.3 (nearly random disper-

sion). For j¼ 0.1444 and the tissue specimen in this case

study, the anisotropy is initially 25 and was predicted to

increase nonlinearly with k. At k¼ 1.23 the ratio Et/Gt

reaches a value of 159. For the assumed value of j¼ 0.01,

the anisotropy was also set to 25 initially, but subsequently

increased more substantially, reaching 2162 at k¼ 1.23.

For the nearly isotropic case j¼ 0.3, the anisotropy was

set to 3 in the unstretched state but increased to merely

31 at k¼ 1.23. The stress-stretch curves and the longitudi-

nal tangent stiffness values were nearly identical for all

three cases. Therefore, the variations in the anisotropy

between j¼ 0.01, j¼ 0.1444, and j¼ 0.3 were due to

drastically different responses in the tangent shear

moduli, Fig. 4(b). For j¼ 0.01, Gt was essentially constant

until k¼ 1.15; yet when the fiber dispersion is accounted

for Gt increases exponentially even from k¼ 1.0. The

impact this nonlinearity and evolution of the anisotropy

with applied stretch has on the dynamic beam models is

examined next.

The first three natural frequencies of vibration in de-

pendence of the longitudinal stretch for the vocal ligament

specimen computed using the Euler–Bernoulli and the

Timoshenko beam models [Eqs. (11) and (13)] are dis-

played in Fig. 5. The first three modes of vibration were

studied because these have been projected to account for

99% of the energy at the onset of self-oscillation.39 The

ligament specimen had an initial length and diameter of

L0¼ 17.12 mm and D0¼ 2.78 mm, respectively, and was

assumed to have a density of qt¼ 1040 kg/m3. In the

unstretched state, the fundamental frequency was 63 and

53 Hz for the Euler–Bernoulli and the Timoshenko models,

respectively. Contrary to the beam models, the string

model equation ðF0 ¼ ð1=2LÞffiffiffiffiffiffiffiffiffiffiffiffiffiffir11=qt

pÞ would predict

F0¼ 0 Hz in the unstretched state. The FTn � k curves were

essentially linear in the small stretch region (k< 1.10) but

then increased strongly beyond k¼ 1.15. This increase of

the frequency for large stretch domain was due to exponen-

tial stiffening of the beam (i.e., ligament) as indicated in

Fig. 3(a). The percent difference between FEBn and FT

n (for

n¼ 1, 2, 3) as a function of the longitudinal stretch is given

in Fig. 5(d). For the first mode, the percent difference was

initially 18% in the undeformed state. As longitudinal

stretch was applied, the percent difference decreased until

k¼ 1.05, but then began increasing monotonically yet was

still within 10% difference. Comparable trends were noted

for modes two and three; however, the percent difference

values were much larger. In the unstretched state of mode

three, the Euler–Bernoulli model estimated a frequency

more than double that of the Timoshenko model. There-

fore, the effects of transverse shear deformation and rotary

inertia had an even greater influence on the natural fre-

quencies at higher modes, similar to other studies.40

The natural frequency predictions are compared for the

two extreme microstructural cases (j¼ 0.01 and j¼ 0.3) in

Fig. 6. Since the case of j¼ 0.3 has a low level of anisot-

ropy, the shear effects are inhibited which causes the differ-

ences between Euler–Bernoulli and Timoshenko models to

FIG. 4. The anisotropy and tangent shear modulus predicted from the GOH

model for three levels of fiber dispersion (j¼ 0.01, j¼ 0.1444, and

j¼ 0.3).



be minimal, particularly for the fundamental mode

[Fig. 6(e)].

IV. DISCUSSION

This investigation presented an approach connecting the

microstructure of the vocal ligament to its biomechanical

characteristics and functional role during phonation. The

microstructural arrangement of the extracellular matrix fibrous

proteins, identified by the inherent crimp to be primarily colla-

gen fibers, was quantified via a fiber dispersion coefficient.

This particular microstructure was subsequently correlated to

the macro-scale biomechanical response under uniaxial ten-

sion. Once the biomechanical properties were determined,

dynamic beam models were developed to gain insights into

phonation processes, in our case vocal fold posturing during a

glissando. Consequently, a novel structure-property-function

technique was illustrated for a vocal ligament specimen.

The biomechanical response was modeled by the non-

linear, anisotropic Gasser–Ogden–Holzapfel constitutive

model. The nonlinearity of the stress-stretch curve at large

deformations was predominantly due to increased contribu-

tion of the fibrous proteins. In the unstretched state, the fibers

were dispersed about the anterior-posterior axis through

the definition of the fiber dispersion coefficient j which

was measured from the SHG image. As the tensile stretch

was applied, the fibers began to align further along the

longitudinal axis which caused the exponential growth in the

stress-stretch response and stiffness at large stretch levels.

Furthermore, the evolution of the anisotropy as tension was

applied was also a direct outcome of the GOH model. This

result could not be replicated in phenomenological models

that do not account for the changing microstructure.

There are distinct advantages of the microstructurally

based constitutive model as opposed to a macro-scale aniso-

tropic constitutive model (e.g., Fung-type hyperelastic

model). First, a macro-scale model can be prone to mathe-

matical deficiencies such as non-convexity when fitting pa-

rameters, where convexity implies that the stiffness tensor

(i.e., second derivative of the strain-energy potential with

respect to e) is positive definite.41 Besides the mathematical

advantages of material stability, a microstructural model is

favorable regarding the biomechanics as well. A macro-scale

anisotropic model can be approximated with the current

FIG. 5. Euler–Bernoulli and Timoshenko beam model predictions of the natural frequencies for the GOH constitutive model and j¼ 0.1444, with the pitch

displayed on the secondary vertical axis.



model by setting j¼ 0.01 (approximating perfectly aligned

fibers). This would be the scenario if one was to use a typical

transversely isotropic model of the vocal folds. While the

longitudinal properties (i.e., tangent Young’s modulus) from

the tensile test could be properly modeled with a macro-

scale anisotropic model, the macro-scale model is incapable

of replicating other physical characteristics such as the evo-

lution of the shear modulus [Fig. 4(b)]. A microstructural

model accounting for the fiber dispersion is critical to accu-

rately model the biomechanical properties in multiple direc-

tions. Complex loading configurations occur in vivo during

phonation, where longitudinal tension is present with shear

deformation. In other biological systems, such as arterial tis-

sues of the cardiovascular system, multi-axial loading condi-

tions are present. This study suggests that a macro-scale

anisotropic constitutive model would be unable to accurately

model the multi-axial behavior. Hence, a microstructural

model accounting for fiber dispersion is obligatory for func-

tional biomechanics. Furthermore, this emphasizes the need

to experimentally measure the fiber dispersion and not sim-

ply fit the parameter. Considering j¼ 0.01, j¼ 0.1444, and

j¼ 0.3 in the analysis above, the curve fits to rN� k yielded

a nearly identical R2 values, making it impossible to distin-

guish between j values. Thus, knowledge of j is essential to

determine the evolution of the tissue anisotropy.

One limitation of the current approach was the assump-

tion that the fiber dispersion j computed from the SHG

image at the mid-membranous (i.e., mid-coronal) location

was uniform throughout the vocal ligament specimen. We

speculate that the microstructural arrangement likely varies

based upon the anterior-posterior location along the vocal

ligament as indicated by the heterogeneous elastic response

measured in previous studies.22,30 A more complete analysis

should include the fiber dispersion at several locations along

the vocal ligament to account for the histological heteroge-

neity. A recent study has documented the dispersion and

density of collagenous fibers of the vocal ligament with

more samples and provided a statistical range.37 Another li-

mitation was that only one specimen was analyzed since this

was a case study. The anatomical and biomechanical varia-

tions in the vocal fold lamina propria will likely lead to a dis-

tinct vibrational frequency paradigm for each subject.

Nevertheless, the specimen analyzed in the present case

study represents a typical male vocal ligament in terms of

geometry (e.g., length and thickness) and mechanical

properties.

FIG. 6. Euler–Bernoulli and Timoshenko beam model predictions for the extreme cases of near perfectly aligned fibers (j¼ 0.01) and near isotropy (j¼ 0.3).

(d) and (e) show the percent difference of the frequency predictions from the Euler–Bernoulli and Timoshenko beam models for j¼ 0.01 and j¼ 0.3,

respectively.



The continuum dynamic beam models of vocal ligament

vibration described above are attractive due to their simplic-

ity. The number of required parameters is small (8 for this

investigation) compared to lumped mass models, which can

involve many parameters (e.g., 28 parameters in Tokuda

et al.42). Euler–Bernoulli (i.e., classical) beam theory is typi-

cally intended for long slender structures (length-to-thick-

ness ratio> 100), and it assumes isotropy. The shear mode

of deformation is known to be a primary factor contributing

to the mucosal wave of vocal fold vibration.43 Yet this mode

cannot be captured in an Euler–Bernoulli beam model (or a

string model) of phonation. This can be remedied by

employing Timoshenko beam theory. A Timoshenko beam

accounts for shear deformation, which is especially impor-

tant for highly anisotropic structures, and is applicable for

thick beams, which seems to be relevant for the vocal liga-

ment (length-to-thickness ratio � 15). In the Timoshenko

beam model of Eq. (13), the role of bending resistance and

rotary inertia only (i.e., Rayleigh beam theory) can be eli-

cited by letting Gt approach infinity, which suppresses shear

deformation. The F0-k curve of this model is within 1% of

the Euler–Bernoulli prediction. Conversely, the role of bend-

ing resistance and shear deformation only (i.e., Shear beam

theory) can be seen by setting the first I/AL2 term of b to

zero, which suppresses rotary inertia. The F0-k curve of this

model is virtually identical to the Timoshenko model in Fig.

5(a), with the differences being less than 1.5%. Therefore,

one can deduce that the major difference between the Euler–

Bernoulli and Timoshenko beam models is the inclusion of

shear deformation not rotary inertia.

In the unstretched state, which is more applicable to nor-

mal speaking conditions, the Euler–Bernoulli model pre-

dicted a fundamental frequency that was 20% greater than

the Timoshenko model. However, this study suggests that

there may not be a drastic difference between FEB0 and FT

0 as

the vocal ligament is elongated. The difference is much

larger at higher modes of vibration. The importance of trans-

verse shear deformation and rotary inertia are more pro-

nounced on the second and third natural frequencies because

the effective “beam” length is shortened in these higher

mode shapes and shear deformation has more influence in

shorter beams. During singing (e.g., a glissando), these

higher modes may play an even larger role than in normal

speaking conditions. Thus, Timoshenko beam theory should

be used when studying vocal fold vibration, especially the

higher modes of vibration.

There are several recommendations for future studies.

First, frequency rise in this investigation was modeled only

as vibration of the passive elastic properties of the vocal lig-

ament. Biological tissues commonly exhibit a hysteretic

response, and the vocal ligament of this case study does so

too, as indicated in Fig. 3. The viscous tissue response under-

pinning the hysteresis is not accounted for presently and

model predictions are based on the centerline of the hystere-

sis loop of the stress-stretch data. Tissue characterization

and constitutive modeling accounting for the anisotropy in

the viscous tissue response is currently ongoing work. Also,

the active properties of the vocalis (thyroarytenoid) muscle

should be incorporated along with the longitudinal tension

since vocalis muscle contraction would stiffen the founda-

tion of the vocal fold, which could be modeled with a Win-

kler foundation as developed in Nielsen.44 Lastly, the

microscopy technique detailed in this investigation could be

used to test the premise that the vocal ligament is trans-

versely isotropic. Most models of the vocal fold lamina prop-

ria that account for tissue anisotropy, suppose the tissue to

be transversely isotropic (e.g., Ref. 45). Acquiring three

dimensional multiphoton images of the vocal fold micro-

structure, as in Miri et al.,25 could allow one to assess if the

fibrous proteins do truly possess rotational symmetry about

the anterior-posterior axis. If this rotational symmetry is

absent then more advanced material models may be

required.

V. CONCLUSION

The present work contributes two primary aspects to

the understanding of phonation: (1) a microstructurally

informed, nonlinear, anisotropic hyperelastic constitutive

model was employed to characterize the biomechanics of

vocal fold tissue, and (2) the effects of the anisotropy in

mechanical properties on the natural frequency of

vibration during vocal ligament elongation were evaluated.

Therefore, a new understanding of the microstructure-

property-function relationship emerged. The approach was

illustrated as a case study for a vocal ligament specimen.

The constitutive model integrated information about the

ligament’s micro-architecture, specifically the dispersion of

collagen fibers. The GOH model allowed the uniaxial test

data to be fit independent of the microstructure characteris-

tics. However, the model became uniquely defined once

data from the transverse indentation and the microstructure

analysis was added. Only with such a complete set of infor-

mation provided is it possible to appropriately describe the

evolution of the vocal ligament anisotropy with posturing.

Such knowledge is relevant in the context of frequency

control. In order to investigate effects of shear deformation

on phonation, the fundamental frequency predictions of an

Euler–Bernoulli beam model and a Timoshenko beam

model of vocal ligament vibration were compared. Not

unexpectedly, the Timoshenko beam model predicted lower

frequencies than the Euler–Bernoulli model. The magni-

tude of the influence of shear deformation on vocal liga-

ment vibration increased with the mode number, and was

found to be strongly dependent on the elongation as well as

the microstructure as reflected in the anisotropy. Shear de-

formation on vocal ligament vibration was predicted to be

most significant in the unstretched state, and subsequently

decline since elongation increases the axial stress in the

vocal ligament. A minimum of shear influence on vibration

was found to occur at stretches 1.05 to 1.15 depending on

the microstructure. At high stretches, shear again becomes

important as the anisotropy of the tissue significantly

increases.

ACKNOWLEDGMENTS

The authors are grateful to the National Institutes of

Health (NIDCD Grant No. R01 DC006101) for funding this



investigation. J. E. K. is thankful to the National Science

Foundation for support in the form of a graduate research

fellowship. Additionally, the authors would like to

acknowledge the assistance of the Live Cell Imaging

Facility at the University of Texas Southwestern Medical

Center.

APPENDIX A: INDENTATION MODEL

Details of the anisotropic contact model used to charac-

terize the transverse indentation experiment are provided

here. The indentation modulus M is

M ¼ 2pðp

0

a3iB�1ij ðcÞa3j

½ða1=a2Þcos2cþ ða2=a1Þsin2c�1=2dc

; (A1)

where a3i are the direction cosines of the indentation

load normal to the surface, B is a Barnett–Lothe tensor, a1

and a2 are the semiaxes of the contact ellipse, and c is

the angle between a vector t and a coordinate axis lying in

the plane of the contact surface.29 The Barnett–Lothe tensor

B is

BðtÞ ¼ �1

p

ðp

0

½ðmnÞ ðnnÞ�1ðnmÞ � ðmmÞ�du; (A2)

with u being the angle between a unit vector m and an arbi-

trary datum in the plane normal to t. The second-order tensors

(ab) are defined as (ab)jk¼ aiCijklbl, where Cijkl is the elastic

stiffness tensor. The unit vectors with respect to the (x, y, z)

coordinate system are m¼ [�cosu sinc cosu cosc sinu]T and

n¼ [sinu sinc� sinu cosc cosu]T, where T denotes the

transpose. Therefore, fitting the indentation force-

displacement response to Eq. (1) allows one to calculate

M and to resolve the individual components of the elastic

stiffness tensor Cijkl.

APPENDIX B: GOH MODEL

The Gasser–Ogden–Holzapfel24 model is solved for the

case of uniaxial tensile stretch k and longitudinal shear strain

c. For stretch k applied in the anterior-posterior direction

(treated as the x-axis) the unimodular (distortional) portion

of the deformation gradient, F is

F5k 0 c0 1=

ffiffiffikp

0

0 0 1=ffiffiffikp

24

35 (B1)

such that the material is incompressible [i.e., det(F)¼ 1].

The longitudinal shear strain is included in F so an analytical

expression of the shear modulus as a function of k may be

obtained from the final solution. The actual value of c will

be set to zero so that no shear strain is present in the model.

The left Cauchy–Green deformation tensor is

b ¼ FFT ¼

k2 þ c2 0 c=ffiffiffikp

0 1=k 0

c=ffiffiffikp

0 1=k

24

35: (B2)

The unit vector defining the preferred fiber orientation is a0,

which is assumed to be aligned along the x-axis. The current

spatial fiber orientation is

a ¼ Fa0 ¼k 0 c0 1=

ffiffiffikp

0

0 0 1=ffiffiffikp

8<:

9=;

1

0

0

8<:

9=; ¼

k0

0

8<:

9=;:

(B3)

The unimodular (i.e., incompressible) structure tensor is

h ¼ jb þ ð1� 3jÞða � aÞ

¼k2ð1� 2jÞ þ jc2 0 jc=

ffiffiffikp

0 j=k 0

jc=ffiffiffikp

0 j=k

24

35; (B4)

where � denotes the outer product or tensor product of two

vectors. The Green–Lagrange strain-like quantity e is

e ¼ tr ðhÞ � 1 ¼ k2ð1� 2jÞ þ jc2 þ 2jk� 1: (B5)

The anisotropic hyperelastic potential of the vocal ligament

is assumed to be composed of non-collagenous ground sub-

stances (matrix) and one family of collagen fibers. The ma-

trix (denoted with subscript “g”) is represented with an

isotropic neo-Hookean model, and the family of collagen

fibers (denoted with subscript “f”) is defined with an expo-

nential scalar stress function wf ¼ k1eexpðk2e2Þ. Thus, the

isochoric Kirchhoff stress tensor s (which is equivalent to

the Cauchy stress tensor r for incompressible materials), is

a superposition of the ground substance and fiber stresses.

The fictitious Kirchhoff stress tensor ~s is computed first

and subsequently transformed into the true Kirchhoff

stress tensor s (or r) via the fourth-order projection tensor

P ¼ I� ð1=3ÞI � I (or in indicial notation Pijkl ¼ ð1=2Þ½dikdjl þ dildjk� � ð1=3ÞdijdklÞ: The fictitious Kirchhoff stress

tensor is

~s ¼ ~sg þ ~sf ¼ cb þ 2wf h ¼cðk2 þ c2Þ þ 2wf ½k2ð1� 2jÞ þ jc2� 0 ðcþ 2wf jÞc=

ffiffiffikp

0 ðcþ 2wf jÞ=k 0

ðcþ 2wf jÞ c=ffiffiffikp

0 ðcþ 2wf jÞ=k

264

375; (B6)

where c is the neo-Hookean parameter. The true Kirchhoff stress s ¼ P : ~s (with: being the double-dot product) is



s ¼ r ¼ 1

3

2~s11 � 2~s22 0 3~s13

0 ~s22 � ~s11 0

3~s13 0 ~s22 � ~s11

24

35

(B7)

and ~s33 is eliminated from the result above because

~s22 ¼ ~s33.

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the anisotropic hyperelastic biomechanical response of the vocal

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